Some Examples of Static Equilibrium — Question 1

Question 1

A ladder load is limited by friction before reaching the top

A uniform $6\,\mathrm m$ ladder of mass $20\,\mathrm{kg}$ leans at $60^\circ$ above horizontal against a smooth wall. The floor has $\mu_s=.40$. A $60\,\mathrm{kg}$ person stands distance $s$ along the ladder from its foot. Use $g=10$.

1. Find the largest admissible $s$ before slipping.

2. Find floor and wall forces at that limit and check that the person is still on the ladder.

Question 1 - Solution

A ladder load is limited by friction before reaching the top

(a) Derivation and calculation

Let $H$ be the wall reaction and $f$ the floor friction. Force balance gives $$N=(20+60)10=800\ \mathrm N,\qquad f=H.$$ At impending slip, $H=\mu_sN=.40(800)=320$ N. Moments about the foot give $$HL\sin60^\circ=\left(200\frac L2+600s\right)\cos60^\circ.$$

(b) Result and interpretation

Solving with $L=6.0$ m, $$s_{\max}=\frac{320(6)\tan60^\circ-200(3)}{600} =\boxed{4.54\ \mathrm m}.$$ The limiting reactions are $$\boxed{N=800\ \mathrm N},\qquad\boxed{H=f=320\ \mathrm N}.$$ Since $s_{\max}<L$, slip limits the climb before the top is reached.

Physical check

The floor friction opposes outward foot motion and the smooth wall supplies only a horizontal reaction.